A ladder, 5.0 m long, leans against a frictionless wall at a
point 4.0 m above the ground. A painter is climbing up the
ladder. The mass of the ladder is 12.0 kg and the mass of the
painter is 60.0 kg.
The ladder begins to slip at its base when the painter is 70 %
of the way up the length of the ladder. What is the coefficient
of static friction between the ladder and the floor?
(The base of the ladder is 3 meters from the wall)
To find the coefficient of static friction between the ladder and the floor, we can analyze the forces acting on the ladder.
Let's consider the forces acting on the ladder: the weight of the ladder (Wladder), the weight of the painter (Wpainter), the normal force (N) exerted by the floor on the ladder, and the static friction force (Ff) between the ladder and the floor.
1. Calculate the weights of the ladder and the painter:
Wladder = mass of ladder × gravitational acceleration
= 12.0 kg × 9.8 m/s^2
Wpainter = mass of painter × gravitational acceleration
= 60.0 kg × 9.8 m/s^2
2. Determine the total horizontal force acting on the ladder:
The total horizontal force acting on the ladder is the sum of the horizontal component of the normal force (N) and the horizontal component of the weight of the painter (Wpainter).
The horizontal component of the normal force is N × cos(θ), where θ is the angle of inclination of the ladder with the horizontal.
We can find the angle of inclination using the given information: the base of the ladder is 3 meters from the wall, and the ladder is 5 meters long.
tan(θ) = height of the ladder / distance from the wall
tan(θ) = 4.0 m / 3.0 m
θ = arctan(4.0 / 3.0)
The horizontal component of the weight of the painter is Wpainter × cos(θ).
Therefore, the total horizontal force is:
Fhorizontal = N × cos(θ) + Wpainter × cos(θ)
3. Determine the maximum value of the static friction force:
The maximum value of the static friction force is equal to the static friction coefficient (μ) multiplied by the normal force (N).
Ffmax = μN
4. Determine the static friction coefficient (μ):
Since the ladder begins to slip at its base when the painter is 70% of the way up the length (or 70% of 5.0 m) of the ladder, we can calculate the height at which the painter is located:
Height of the painter = 70% × 5.0 m
To find the normal force (N) exerted by the floor on the ladder, we can use the vertical force balance equation:
N + Wladder + Wpainter = 0
N = -Wladder - Wpainter
N = -(Wladder + Wpainter)
The static friction force must be equal to the total horizontal force:
Ff = Fhorizontal
Therefore:
μN = N × cos(θ) + Wpainter × cos(θ)
μ = (N × cos(θ) + Wpainter × cos(θ)) / N
Substitute the value of N and solve for μ.
Note: In this problem, the static friction force is at its maximum value because the ladder is about to slip.